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Singular surfaces, mod 2 homology, and hyperbolic volume, I


Authors: Ian Agol, Marc Culler and Peter B. Shalen
Journal: Trans. Amer. Math. Soc. 362 (2010), 3463-3498
MSC (2000): Primary 57M50
DOI: https://doi.org/10.1090/S0002-9947-10-04362-X
Published electronically: February 2, 2010
MathSciNet review: 2601597
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Abstract | References | Similar Articles | Additional Information

Abstract: If $ M$ is a simple, closed, orientable $ 3$-manifold such that $ \pi_1(M)$ contains a genus-$ g$ surface group, and if $ H_1(M;\mathbb{Z}_2)$ has rank at least $ 4g-1$, we show that $ M$ contains an embedded closed incompressible surface of genus at most $ g$. As an application we show that if $ M$ is a closed orientable hyperbolic $ 3$-manifold of volume at most $ 3.08$, then the rank of $ H_1(M;\mathbb{Z}_2)$ is at most $ 6$.


References [Enhancements On Off] (What's this?)

  • 1. Ian Agol, Tameness of hyperbolic $ 3$-manifolds, arXiv:math/0405568v1[math.GT].
  • 2. Ian Agol, Peter A. Storm, and William P. Thurston, Lower bounds on volumes of hyperbolic Haken $ 3$-manifolds, J. Amer. Math. Soc. 20 (2007), no. 4, 1053-1077 (electronic). With an appendix by Nathan Dunfield. MR 2328715
  • 3. James W. Anderson, Richard D. Canary, Marc Culler, and Peter B. Shalen, Free Kleinian groups and volumes of hyperbolic $ 3$-manifolds, J. Differential Geom. 43 (1996), no. 4, 738-782. MR 1412683 (98c:57012)
  • 4. Danny Calegari and David Gabai, Shrinkwrapping and the taming of hyperbolic $ 3$-manifolds, J. Amer. Math. Soc. 19 (2006), no. 2, 385-446 (electronic). MR 2188131 (2006g:57030)
  • 5. Marc Culler and Peter B. Shalen, Singular surfaces, mod $ 2$ homology, and hyperbolic volume, II, arXiv:math/070166v5[math.GT].
  • 6. Marc Culler and Peter B. Shalen, Paradoxical decompositions, $ 2$-generator Kleinian groups, and volumes of hyperbolic $ 3$-manifolds, J. Amer. Math. Soc. 5 (1992), no. 2, 231-288. MR 1135928 (93a:57017)
  • 7. David Gabai, Foliations and the topology of $ 3$-manifolds, J. Differential Geom. 18 (1983), no. 3, 445-503. MR 723813 (86a:57009)
  • 8. John Hempel, $ 3$-manifolds, AMS Chelsea Publishing, Providence, RI, 2004. Reprint of the 1976 original. MR 2098385 (2005e:57053)
  • 9. William Jaco and Peter B. Shalen, Peripheral structure of $ 3$-manifolds, Invent. Math. 38 (1976), no. 1, 55-87. MR 0428332 (55:1357)
  • 10. William Meeks, III, Leon Simon, and Shing Tung Yau, Embedded minimal surfaces, exotic spheres, and manifolds with positive Ricci curvature, Ann. of Math. (2) 116 (1982), no. 3, 621-659. MR 678484 (84f:53053)
  • 11. C. D. Papakyriakopoulos, On Dehn's lemma and the asphericity of knots, Ann. of Math. (2) 66 (1957), 1-26. MR 0090053 (19:761a)
  • 12. Grisha Perelman, Ricci flow with surgery on three-manifolds, arXiv:math/ 0303109v1[math.DG].
  • 13. Peter B. Shalen and Philip Wagreich, Growth rates, $ Z\sb p$-homology, and volumes of hyperbolic $ 3$-manifolds, Trans. Amer. Math. Soc. 331 (1992), no. 2, 895-917. MR 1156298 (93d:57002)
  • 14. Arnold Shapiro and J. H. C. Whitehead, A proof and extension of Dehn's lemma, Bull. Amer. Math. Soc. 64 (1958), 174-178. MR 0103474 (21:2242)
  • 15. Jonathan Simon, Compactification of covering spaces of compact $ 3$-manifolds, Michigan Math. J. 23 (1976), no. 3, 245-256 (1977). MR 0431176 (55:4178)
  • 16. John Stallings, On the loop theorem, Ann. of Math. (2) 72 (1960), 12-19. MR 0121796 (22:12526)
  • 17. -, Homology and central series of groups, J. Algebra 2 (1965), 170-181. MR 0175956 (31:232)
  • 18. Friedhelm Waldhausen, On irreducible $ 3$-manifolds which are sufficiently large, Ann. of Math. (2) 87 (1968), 56-88. MR 0224099 (36:7146)
  • 19. Jeff Weeks and Craig Hodgson, ClosedCensusInvariants.txt, downloadable text file.

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Additional Information

Ian Agol
Affiliation: Department of Mathematics, University of California, Berkeley, 970 Evans Hall, Berkeley, California 94720-3840
Email: ianagol@math.berkeley.edu

Marc Culler
Affiliation: Department of Mathematics, Statistics, and Computer Science (M/C 249), University of Illinois at Chicago, 851 S. Morgan St., Chicago, Illinois 60607-7045
Email: culler@math.uic.edu

Peter B. Shalen
Affiliation: Department of Mathematics, Statistics, and Computer Science (M/C 249), University of Illinois at Chicago, 851 S. Morgan St., Chicago, Illinois 60607-7045
Email: shalen@math.uic.edu

DOI: https://doi.org/10.1090/S0002-9947-10-04362-X
Received by editor(s): July 3, 2005
Received by editor(s) in revised form: February 2, 2008
Published electronically: February 2, 2010
Additional Notes: This work was partially supported by NSF grants DMS-0204142 and DMS-0504975
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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